Finite Group Databasebeta

← #12027700

Subgroups — #12027700

Order 1280.

2018 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#390C211yesmaximal, characteristic
2C2×#65509C212yesmaximal
3C2×#65239C212yesmaximal, complemented
4#10602358C212yesmaximal, complemented
5#10602339C212yesmaximal, complemented
6#10602339C212yesmaximal, complemented
7#10602337C212yesmaximal, complemented
8#10602328C211yesmaximal, characteristic, complemented
9#10602327C211yesmaximal, characteristic, complemented
10C23×C5×D4C2211yescharacteristic
11C24×Dic5C2212yes—
12C22×C5×C4⋊C4C2211yescharacteristic
13C22×C5×C22⋊C4C2212yes—
14C22×C5×C22⋊C4C2212yes—
15C22×C42×C5C2211yescharacteristic
16C22×C22⋊Dic5C2212yes—
17C22×C22⋊Dic5C2212yes—
18C22×C22⋊Dic5C2212yes—
19C22×C4⋊Dic5C2212yes—
20C22×C4⋊Dic5C2212yes—
21C5×#390C418yescomplemented
22#65509—28no—
23C2×#3963C2212yes—
24C2×#3963C2211yescharacteristic
25C2×#3963C2212yes—
26C2×#3963C2212yes—
27C2×#3963C2212yes—
28C2×#3963C2211yescharacteristic
29C2×#3963C2211yescharacteristic
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2211yescharacteristic
33#65239—28no—
34#14438—51nomaximal, Sylow
35C25×C5C2312yes—
36C22×C5×D4—42no—
37C22×C5×D4—42no—
38C22×C5×D4C2×C414yes—
39C22×C5×D4D414yes—
40C22×C5×D4Q814yes—
41C23×C4×C5C2312yes—
42C23×C4×C5C2311yescharacteristic
43C23×C4×C5C2311yescharacteristic
44C23×C4×C5C2311yescharacteristic
45C23×Dic5—24no—
46C23×Dic5—24no—
47C23×Dic5—24no—
48C23×Dic5—42no—
49C23×Dic5—42no—
50C23×Dic5—42no—
51C23×Dic5—42no—
52C23×Dic5C2312yes—
53C23×Dic5C2312yes—
54C23×Dic5C2312yes—
55C23×Dic5C2312yes—
56C2×C5×C4⋊C4—42no—
57C2×C5×C4⋊C4C2×C414yescomplemented
58C2×C5×C22⋊C4—24no—
59C2×C5×C22⋊C4—24no—
60C2×C5×C22⋊C4—42no—
61C2×C5×C22⋊C4—42no—
62C2×C5×C22⋊C4C2×C418yescomplemented
63C2×C5×C22⋊C4C2×C418yescomplemented
64C2×C42×C5—42no—
65C2×C42×C5C2×C414yescomplemented
66C2×C22⋊Dic5—24no—
67C2×C22⋊Dic5—24no—
68C2×C22⋊Dic5—24no—
69C2×C22⋊Dic5—24no—
70C2×C22⋊Dic5—24no—
71C2×C22⋊Dic5—42no—
72C2×C22⋊Dic5—42no—
73C2×C22⋊Dic5—42no—
74C2×C22⋊Dic5—42no—
75C2×C4⋊Dic5—24no—
76C2×C4⋊Dic5—24no—
77C2×C4⋊Dic5—24no—
78C2×C4⋊Dic5—42no—
79C2×C4⋊Dic5—42no—
80C2×C4⋊Dic5—42no—
81#3963—24no—
82#3963—24no—
83#3963—22no—
84#3963—24no—
85#3963—41no—
86#3963—42no—
87#3963—42no—
88#3963—41no—
89#3963—42no—
90#3963D414yescomplemented
91C2×#390D511yescharacteristic
92C2×#386—52no—
93C2×#385—52no—
94#1441—52no—
95#1439—52no—
96#1439—52no—
97C4⋊#95—51no—
98#1434—51no—
99C22⋊#95—52no—
100C24×C5—22no—
101C24×C5—22no—
102C24×C5—22no—
103C24×C5—22no—
104C24×C5—22no—
105C24×C5—22no—
106C24×C5—22no—
107C24×C5—22no—
108C24×C5—22no—
109C24×C5—22no—
110C24×C5—22no—
111C24×C5—22no—
112C24×C5C22×C414yes—
113C24×C5C2×D414yes—
114C24×C5C2×Q814yes—
115C24×C5C2411yescharacteristic
116C2×C5×D4—44no—
117C2×C5×D4—44no—
118C2×C5×D4—48no—
119C2×C5×D4—48no—
120C2×C5×D4C4⋊C4116yescomplemented
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—22no—
131C22×C4×C5—22no—
132C22×C4×C5—22no—
133C22×C4×C5—21no—
134C22×C4×C5—21no—
135C22×C4×C5C22×C412yes—
136C22×C4×C5C22×C414yes—
137C22×C4×C5C22×C412yes—
138C22×C4×C5C22×C412yes—
139C22×C4×C5C2×D412yes—
140C22×C4×C5C2×D412yes—
141C22×C4×C5C2×Q812yes—
142C22×C4×C5#4112yes—
143C22×Dic5—22no—
144C22×Dic5—22no—
145C22×Dic5—22no—
146C22×Dic5—22no—
147C22×Dic5—22no—
148C22×Dic5—22no—
149C22×Dic5—22no—
150C22×Dic5—22no—
151C22×Dic5—22no—
152C22×Dic5—22no—
153C22×Dic5—22no—
154C22×Dic5—22no—
155C22×Dic5—22no—
156C22×Dic5—22no—
157C22×Dic5—22no—
158C22×Dic5—22no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5—22no—
162C22×Dic5—22no—
163C22×Dic5—22no—
164C22×Dic5—22no—
165C22×Dic5—22no—
166C22×Dic5—22no—
167C22×Dic5—22no—
168C22×Dic5—22no—
169C22×Dic5—48no—
170C22×Dic5—48no—
171C22×Dic5—48no—
172C22×Dic5—48no—
173C22×Dic5—48no—
174C22×Dic5—48no—
175C22×Dic5—48no—
176C22×Dic5C2×D414yes—
177C22×Dic5#4114yes—
178C5×C4⋊C4—44no—
179C5×C22⋊C4—216no—
180C5×C22⋊C4—48no—
181C42×C5—44no—
182C22⋊Dic5—48no—
183C22⋊Dic5—48no—
184C22⋊Dic5—48no—
185C4⋊Dic5—48no—
186C4⋊Dic5—48no—
187C23×D4C2×D511yescharacteristic
188C24×C4—52no—
189C22×C4⋊C4—52no—
190C22×C4⋊C4—52no—
191C22×C4⋊C4C2×D511yescharacteristic
192C22×C22⋊C4—52no—
193C22×C22⋊C4—52no—
194C22×C22⋊C4—52no—
195C22×C22⋊C4C2×D512yes—
196C22×C22⋊C4C2×D512yes—
197C22×C42C2×D511yescharacteristic
198#390Dic518yescomplemented
199#386—108no—
200#385—108no—
201C2×#95—51no—
202C2×#95—52no—
203C2×#95—51no—
204C2×#95—52no—
205C2×#95—52no—
206C2×#95—51no—
207C2×#95—52no—
208C2×#95—52no—
209C2×#95—51no—
210C2×#95—52no—
211C23×C5—24no—
212C23×C5—24no—
213C23×C5—24no—
214C23×C5—24no—
215C23×C5—24no—
216C23×C5—24no—
217C23×C5—24no—
218C23×C5—24no—
219C23×C5—24no—
220C23×C5—24no—
221C23×C5—24no—
222C23×C5—24no—
223C23×C5—24no—
224C23×C5—24no—
225C23×C5—24no—
226C23×C5—24no—
227C23×C5—24no—
228C23×C5—24no—
229C23×C5—42no—
230C23×C5—42no—
231C23×C5—42no—
232C23×C5—42no—
233C23×C5—42no—
234C23×C5—42no—
235C23×C5—42no—
236C23×C5—42no—
237C23×C5—42no—
238C23×C5—42no—
239C23×C5—42no—
240C23×C5—42no—
241C23×C5—42no—
242C23×C5—42no—
243C23×C5—42no—
244C23×C5—42no—
245C23×C5C2×C4⋊C418yes—
246C23×C5C23×C411yescharacteristic
247C23×C5C22×D411yescharacteristic
248C23×C5C22×D411yescharacteristic
249C23×C5C22×Q811yescharacteristic
250C23×C5C2×#4111yescharacteristic
251C23×C5C2×#4112yes—
252C23×C5H(5,𝔽2)12yes—
253C23×C5H(5,𝔽2)12yes—
254C23×C5H(5,𝔽2)12yes—
255C23×C5H(5,𝔽2)11yescharacteristic
256C23×C52−1+411yescharacteristic
257C5×D4—416no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—22no—
261C2×C4×C5—22no—
262C2×C4×C5—24no—
263C2×C4×C5—24no—
264C2×C4×C5—24no—
265C2×C4×C5—24no—
266C2×C4×C5—24no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—24no—
270C2×C4×C5—24no—
271C2×C4×C5—22no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—42no—
276C2×C4×C5—42no—
277C2×C4×C5—42no—
278C2×C4×C5C2×C4⋊C414yes—
279C2×C4×C5C4×D414yes—
280C2×Dic5—24no—
281C2×Dic5—24no—
282C2×Dic5—24no—
283C2×Dic5—24no—
284C2×Dic5—24no—
285C2×Dic5—24no—
286C2×Dic5—432no—
287C2×Dic5—42no—
288C2×Dic5—42no—
289C2×Dic5—42no—
290C2×Dic5—42no—
291C2×Dic5—42no—
292C2×Dic5—42no—
293C2×Dic5—42no—
294C2×Dic5—42no—
295C2×Dic5—42no—
296C2×Dic5—42no—
297C2×Dic5—42no—
298C2×Dic5—42no—
299C2×Dic5—42no—
300C2×Dic5—42no—
301C2×Dic5—42no—
302C2×Dic5—42no—
303C2×Dic5—42no—
304C2×Dic5—42no—
305C2×Dic5—42no—
306C2×Dic5—42no—
307C2×Dic5—42no—
308C2×Dic5—42no—
309C25C22×D512yes—
310C22×D4—42no—
311C22×D4—42no—
312C22×D4Dic1014yes—
313C22×D4D2014yes—
314C22×D4C2×Dic514yes—
315C23×C4—52no—
316C23×C4—52no—
317C23×C4—52no—
318C23×C4—52no—
319C23×C4—104no—
320C23×C4—104no—
321C23×C4—104no—
322C23×C4—202no—
323C23×C4—202no—
324C23×C4—202no—
325C23×C4—202no—
326C23×C4C22×D512yes—
327C23×C4C22×D511yescharacteristic
328C23×C4C22×D511yescharacteristic
329C23×C4C22×D511yescharacteristic
330C2×C4⋊C4—42no—
331C2×C4⋊C4—104no—
332C2×C4⋊C4—104no—
333C2×C4⋊C4—104no—
334C2×C4⋊C4—202no—
335C2×C4⋊C4—202no—
336C2×C4⋊C4—202no—
337C2×C4⋊C4C2×Dic514yescomplemented
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—42no—
341C2×C22⋊C4—42no—
342C2×C22⋊C4—104no—
343C2×C22⋊C4—104no—
344C2×C22⋊C4—104no—
345C2×C22⋊C4—104no—
346C2×C22⋊C4—104no—
347C2×C22⋊C4—202no—
348C2×C22⋊C4—202no—
349C2×C22⋊C4—202no—
350C2×C22⋊C4—202no—
351C2×C22⋊C4C2×Dic518yescomplemented
352C2×C22⋊C4C2×Dic518yescomplemented
353C2×C42—42no—
354C2×C42C2×Dic514yescomplemented
355#95—54no—
356#95—104no—
357#95—104no—
358#95—104no—
359#95—102no—
360#95—202no—
361#95—202no—
362#95—202no—
363#95—201no—
364#95—201no—
365C22×C5—28no—
366C22×C5—28no—
367C22×C5—28no—
368C22×C5—44no—
369C22×C5—44no—
370C22×C5—44no—
371C22×C5—44no—
372C22×C5—44no—
373C22×C5—44no—
374C22×C5—44no—
375C22×C5—44no—
376C22×C5—44no—
377C22×C5—44no—
378C22×C5—44no—
379C22×C5—44no—
380C22×C5C22×C4⋊C411yescharacteristic
381C22×C5C2×C4×D411yescharacteristic
382C22×C5C4×#4111yescharacteristic
383C22×C5#51811yescharacteristic
384C22×C5#51811yescharacteristic
385C22×C5#51811yescharacteristic
386C22×C5#52011yescharacteristic
387C22×C5#53412yes—
388C22×C5#53512yes—
389C22×C5#53812yes—
390C22×C5#54012yes—
391C22×C5#54212yes—
392C22×C5#54312yes—
393C22×C5D4211yescharacteristic
394C22×C5#54612yes—
395C22×C5#54612yes—
396C22×C5#54711yescharacteristic
397C22×C5D4×Q811yescharacteristic
398C22×C5#55312yes—
399C22×C5#55412yes—
400C22×C5#55411yescharacteristic
401C22×C5#56012yes—
402C22×C5#56212yes—
403C4×C5—24no—
404C4×C5—24no—
405C4×C5—44no—
406C4×C5—42no—
407Dic5—44no—
408Dic5—44no—
409Dic5—82no—
410Dic5—82no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24C2×Dic1014yes—
424C24C2×D2014yes—
425C24C22×Dic514yes—
426C24C23×D511yescharacteristic
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4C4⋊Dic5116yescomplemented
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—54no—
447C22×C4—54no—
448C22×C4—102no—
449C22×C4—102no—
450C22×C4—102no—
451C22×C4—102no—
452C22×C4—102no—
453C22×C4—102no—
454C22×C4—102no—
455C22×C4—102no—
456C22×C4—102no—
457C22×C4—102no—
458C22×C4—102no—
459C22×C4—102no—
460C22×C4—102no—
461C22×C4—102no—
462C22×C4—102no—
463C22×C4—102no—
464C22×C4—102no—
465C22×C4—102no—
466C22×C4—102no—
467C22×C4—102no—
468C22×C4—102no—
469C22×C4—102no—
470C22×C4—102no—
471C22×C4—102no—
472C22×C4—102no—
473C22×C4—102no—
474C22×C4—208no—
475C22×C4—208no—
476C22×C4—208no—
477C22×C4—208no—
478C22×C4—208no—
479C22×C4—208no—
480C22×C4—208no—
481C22×C4C2×Dic1012yes—
482C22×C4C2×D2012yes—
483C22×C4D4×D512yes—
484C22×C4#71112yes—
485C22×C4C22×Dic512yes—
486C22×C4C22×Dic514yes—
487C22×C4C22×Dic512yes—
488C22×C4C22×Dic512yes—
489C4⋊C4—44no—
490C4⋊C4—208no—
491C4⋊C4—208no—
492C22⋊C4—216no—
493C22⋊C4—48no—
494C22⋊C4—208no—
495C22⋊C4—208no—
496C22⋊C4—208no—
497C42—44no—
498C2×C5—48no—
499C2×C5#231511yescharacteristic
500C2×C5#232811yescharacteristic
501C2×C5#233711yescharacteristic
502C2×C5D4×C4⋊C411yescharacteristic
503C2×C5#234911yescharacteristic
504C2×C5#236411yescharacteristic
505C2×C5#237711yescharacteristic
506C2×C5#267812yes—
507C2×C5#269212yes—
508C2×C5#270112yes—
509C2×C5#278012yes—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23C2×C4⋊Dic518yes—
545C23C22×Dic1011yescharacteristic
546C23C22×D2011yescharacteristic
547C23C2×D4×D511yescharacteristic
548C23C2×#71111yescharacteristic
549C23#414412yes—
550C23#414412yes—
551C23#414412yes—
552C23D5×#4112yes—
553C23#414911yescharacteristic
554C23#415011yescharacteristic
555C23C23×Dic511yescharacteristic
556D4—416no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—104no—
578C2×C4—104no—
579C2×C4—104no—
580C2×C4—104no—
581C2×C4—104no—
582C2×C4—104no—
583C2×C4—2032no—
584C2×C4—202no—
585C2×C4—202no—
586C2×C4—202no—
587C2×C4—202no—
588C2×C4—202no—
589C2×C4—202no—
590C2×C4—202no—
591C2×C4—202no—
592C2×C4—202no—
593C2×C4—202no—
594C2×C4—202no—
595C2×C4—202no—
596C2×C4—202no—
597C2×C4—202no—
598C2×C4—202no—
599C2×C4—202no—
600C2×C4—202no—
601C2×C4—202no—
602C2×C4—202no—
603C2×C4—202no—
604C2×C4—202no—
605C2×C4—202no—
606C2×C4C2×C4⋊Dic514yes—
607C2×C4D4×Dic514yes—
608C5#1443811yescharacteristic, complemented, Sylow
609C22—28no—
610C22—28no—
611C22—28no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22#6581712yes—
625C22#6582712yes—
626C22D4×Dic1011yescharacteristic
627C22#6587312yes—
628C22#6587711yescharacteristic
629C22D4×D2011yescharacteristic
630C22#6588812yes—
631C22#6588911yescharacteristic
632C22#6593512yes—
633C22#6594412yes—
634C22#6595412yes—
635C22#6595512yes—
636C22#6600312yes—
637C22#6601212yes—
638C22#6604912yes—
639C22#6605312yes—
640C22C22×C4⋊Dic511yescharacteristic
641C22C2×D4×Dic511yescharacteristic
642C22#6613211yescharacteristic
643C22#6613211yescharacteristic
644C22#6613211yescharacteristic
645C22#41×Dic511yescharacteristic
646C22#6616111yescharacteristic
647C4—24no—
648C4—24no—
649C4—42no—
650C4—44no—
651C4—204no—
652C4—204no—
653C4—402no—
654C4—402no—
655C2—48no—
656C2#1060811312yes—
657C2#1060811712yes—
658C2#1060814412yes—
659C2#1060819412yes—
660C2#1061382511yescharacteristic
661C2D4×C4⋊Dic511yescharacteristic
662C2#1061406411yescharacteristic
663C2#1061436911yescharacteristic
664C2#1061437011yescharacteristic
665C2#1061468511yescharacteristic
666C2#1061490311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C5×#390C2
socle, cosocleC24×C5C24
prime radicalC23×C5×D4C22
hypercenterC2×#390D5
hypocenterC5#14438
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#390

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#65239C2semidirect
4. #10602358C2semidirect
5. #10602339C2semidirect
6. #10602339C2semidirect
7. #10602337C2semidirect
8. #10602328C2semidirect
9. #10602327C2semidirect
21. C5×#390C4semidirect
57. C2×C5×C4⋊C4C2×C4semidirect
62. C2×C5×C22⋊C4C2×C4semidirect
63. C2×C5×C22⋊C4C2×C4semidirect
65. C2×C42×C5C2×C4semidirect
90. #3963D4semidirect
120. C2×C5×D4C4⋊C4semidirect
198. #390Dic5semidirect
337. C2×C4⋊C4C2×Dic5semidirect
351. C2×C22⋊C4C2×Dic5semidirect
352. C2×C22⋊C4C2×Dic5semidirect
354. C2×C42C2×Dic5semidirect
431. C2×D4C4⋊Dic5semidirect
608. C5#14438semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C5×#3902
C2×#655092
C2×#652392
#106023582
#106023392
#106023372
#106023282
#106023272
#144385